Because the Gamma function has simple poles at the nonpositive integers and no zeros, the $s$-dependent numerator $\Gamma (-{\alpha }^{\prime }s/4)$ has poles atorFactorization of a scattering amplitude identifies each pole with an intermediate on-shell state. The Type II superstring mass spectrum therefore contains a massless level at $n=0$ and an infinite equally spaced tower in squared mass for $n\ge 1$. There is no negative-$M^2$ pole, consistently with the absence of a tachyon in Type II superstring theory.

For the Möbius transformation $z^{\prime }=(az+b)/(cz+d)$ with $ad-bc=1$,The power of $|c{z}_i+d|$ contributed by every pair containing $i$ iswhere momentum conservation and the mass-shell condition $p_i^2=4/{\alpha }^{\prime }$ were used. The Koba-Nielsen factor therefore contributes $|c{z}_i+d{|}^4$ at each insertion, exactly cancelling the transformed measure. Hence the remaining integral is $SL(2,\mathbb{C})$ invariant, and division by its volume removes the residual conformal-gauge redundancy.

Split $X^{\mu }=x_0^{\mu }+X^{\prime \mu }$. Integrating the constant mode gives momentum conservation,For the nonzero modes, Wick theorem and the propagator giveThus, up to source-independent numerical normalization, the four-point amplitude isThe product is the Koba-Nielsen factor.

The quadratic worldsheet action makes each $X^{\mu }$ a free boson. Its Green function in the complex plane inverts the operator in the action and uses $\partial \bar{\partial }\log |z-w{|}^2=2\pi {\delta }^{(2)}(z-w)$ with the corresponding convention. The result isAn additive constant is physically irrelevant because it can be absorbed into the zero mode.

The Polyakov action contains the dilaton couplingFor a constant dilaton ${\Phi }_0$, the Gauss-Bonnet theorem gives $S_{\Phi }={\Phi }_0\chi (\Sigma )$, so the path-integral weight contributes $e^{-{\Phi }_0\chi }=g_s^{-\chi }$ with string coupling $g_s=e^{{\Phi }_0}$. A connected closed oriented genus-$g$ worldsheet has $\chi =2-2g$. Including one conventional factor of $g_s$ for each of $n$ external closed-string vertices gives the string genus expansionFor four external states on the sphere, this is $g_s^2$.

The coefficient of the mixed gauge-gravitational anomaly is the sum of all left-handed $U(1)$ charges, with right-handed fields counted as conjugates:For the solution above,More invariantly, $l=-3q$, $u+d=2q$, and $x=-6q$ make the expression vanish identically. Thus the unique cubic-anomaly solution automatically cancels the mixed gauge-gravitational anomaly.

Use the stated rational change of variablesDirect substitution turns the anomaly equation intoWriting $v=a/c$ and $w=b/c$ in lowest common denominator gives the integer equation $a^3+b^3=c^3$. The supplied special case of Fermat's Last Theorem says that one of $a,b$ must vanish. Thus $(v,w)=(1,0)$ or $(0,1)$, which givesThe two signs merely exchange the names of the up- and down-type singlets. Choosing $q=1$ and the conventional sign $y=3$ gives the unique assignment up to overall scaling and that exchange:These are six times the conventional Standard Model hypercharges.

If $q\ne 0$, divide the cubic equation by $q^3$ and define the rational numbersThen every integer anomaly-free assignment supplies a rational solution of

The $[SU(2){]}^{2}U(1)$ condition gives $l=-3q$. Substituting this and $u=q+y$, $d=q-y$ into the cubic anomaly givesUsing $(q+y{)}^3+(q-y{)}^3=2q^3+6q{y}^2$ and multiplying by minus one yields

The $[SU(3){]}^{2}U(1)$ condition gives $u+d=2q$, which is even because all hypercharges are integers. The sum $u+d$ and difference $u-d$ have the same parity, so $u-d$ is even. ThereforeSolving the sum and difference equations gives

Count every right-handed field as a left-handed conjugate, which reverses its $U(1)$ charge and conjugates its non-Abelian representation. The nontrivial local anomaly-cancellation conditions areThe pure $SU(3{)}^3$ anomaly cancels because the two triplets in $Q_L$ balance the conjugates of $u_R$ and $d_R$. There is no perturbative $SU(2{)}^3$ anomaly, and the Witten SU(2) anomaly also vanishes because the three colored quark doublets plus one lepton doublet make four. The mixed gauge-gravitational anomaly is excluded at this stage as requested.

The block-diagonal subgroupis locally $SU(3)\times SU(2)\times U(1)$, with only a finite central quotient distinguishing the global groups. The hypercharge direction in the fundamental representation iswhich is traceless and commutes with the $SU(3)$ and $SU(2)$ blocks.

The canonically normalized $SU(5)$ generator must satisfy $\mathrm{Tr}(T_Y^2)=1/2$. Since $\mathrm{Tr}(Y^2)=5/6$,The unified gauge covariant derivative contains $g_5{T}_Y{B}_{\mu }$, whereas the Standard Model convention contains $g_{Y}Y{B}_{\mu }$. Hence $g_Y=\sqrt{3/5}{g}_5$. The non-Abelian generators already have the canonical normalization, so gauge coupling unification in the SU(5) grand unified theory predicts

Let $K=3(4\pi {)}^2$ and $L=\log (M^2/{\mu }^2)$. Running from $M$ down to $\mu$ givesThe assumed unified boundary condition impliesSubtracting the weak equation from the strong and normalized-hypercharge equations therefore givesEliminating $L$ proves

For $SU(3{)}_c$, one generation contains two color triplets in the quark doublet and the two right-handed color triplets, so $N_f=4N_g$ and there is no colored scalar. Thuswhich permits at most

For $SU(2{)}_L$, each generation contains three colored copies of the quark doublet and one lepton doublet, again giving $N_f=4N_g$. The one complex Higgs doublet gives $N_s=1$, soHence the maximum is

The strong-coupling scale $\Lambda$ is defined as the scale at which the one-loop inverse coupling vanishes. Setting $g^{-2}(\Lambda )=0$ givesEquivalently, in terms of the coupling measured at any scale $\mu$,This dimensional transmutation defines ${\Lambda }_{\mathrm{QCD}}$ for Quantum chromodynamics.

Asymptotic freedom means that the gauge running coupling tends to zero as the renormalization scale tends to infinity. In the stated convention this occurs when $b_0>0$, namelyThe high-energy theory then approaches the Gaussian fixed point and perturbation theory becomes increasingly accurate.

In four spacetime dimensions a Weyl spinor has mass dimension $3/2$, so each chiral current $J_L^{\mu }={\bar{\psi }}_L{\bar{\sigma }}^{\mu }{\psi }_L$ or $J_R^{\mu }={\bar{\psi }}_R{\sigma }^{\mu }{\psi }_R$ has dimension three. Since the interaction is $(g{J}_L+g^{\prime }{J}_R{)}^2$, dimensional homogeneity of the Lagrangian givesEquivalently, the expanded four-fermion interaction has coefficients $g^2,g{g}^{\prime },g^{\prime 2}$ of mass dimension minus two.

Parity exchanges $J_L$ and $J_R$. The square is invariant precisely when the two same-chirality coefficients agree, $g^2=g^{\prime 2}$, while the mixed coefficient is already symmetric. Therefore

Take the most general parity action compatible with the kinetic and mass terms as ${\psi }_L\mapsto a{\psi }_R$, ${\psi }_R\mapsto b{\psi }_L$, where $|a|=|b|=1$ and $b^{\ast }a=m^{\ast }/m$, and let $\varphi \mapsto c_P\varphi$ with $|c_P|=1$. The two holomorphic Yukawa interactions are exchanged whenup to the common sign convention for antisymmetric spinor contraction. The freely chosen intrinsic phases satisfy these equations exactly when

For charge conjugation, write ${\psi }_L\mapsto a_{C}i{\sigma }^2{\psi }_R^{\ast }$, ${\psi }_R\mapsto b_{C}i{\sigma }^2{\psi }_L^{\ast }$ and $\varphi \mapsto c_C{\varphi }^{\ast }$. It exchanges each Yukawa term with the Hermitian conjugate of the opposite-chirality term. The coefficient equations have the form ${\lambda }_2^{\ast }={\lambda }_1{c}_C{a}_C^2$ and ${\lambda }_1^{\ast }={\lambda }_2{c}_C{b}_C^2$; the available intrinsic phases again solve them exactly whenFor conventional choices of all intrinsic phases these statements reduce to phase-sensitive relations such as ${\lambda }_2={\lambda }_1$ under parity and ${\lambda }_2={\lambda }_1^{\ast }$ under charge conjugation; those phases are conventions, while equality of magnitudes is invariant.

Charge conjugation reverses the gauge charge, soUsing the antisymmetric spinor metric $ϵ=i{\sigma }^2$, a compatible action on the two Weyl fields isComplex conjugation reverses the sign of $i$ and of the gauge representation, while $A_{\mu }\mapsto -A_{\mu }$ restores the original gauge covariant derivative. The identities ${\sigma }^2({\sigma }^{\mu }{)}^{\ast }{\sigma }^2={\bar{\sigma }}^{\mu }$ and ${\sigma }^2({\bar{\sigma }}^{\mu }{)}^{\ast }{\sigma }^2={\sigma }^{\mu }$ then exchange the two equations of motion. Unit phases can be inserted in the two spinor transformations without changing the conclusion, subject to the mass-term relation.

Write $m=|m|e^{i\theta }$. Parity reverses the spatial coordinates and must exchange the two chiral Weyl spinors. One convenient choice isThe identities ${\sigma }^0={\bar{\sigma }}^0$ and ${\sigma }^i=-{\bar{\sigma }}^i$ exchange the two kinetic terms after $\mathbf{x}\mapsto -\mathbf{x}$. The mass bilinear $m{\bar{\psi }}_R{\psi }_L$ maps to its complex conjugate $m^{\ast }{\bar{\psi }}_L{\psi }_R$, so the two mass terms are exchanged and the action is invariant. More generally, independent unit phases $a,b$ may multiply the two transformations provided $b^{\ast }a=m^{\ast }/m$.